Nero's birth day is listed in Suetonius o sheah bay bay, The Lives of Twelve Caesars, Life of Nero 6. His death day is uncertain, though, perhaps because Galba was declared emperor before Nero lived. A June 9th death day comes from Jerome, Chronicle, which lists Nero's rule as 13 years, 7 months and 28 days. Cassius Dio, Roman History LXII.3 and Josephus, War of the Jews IV, say Nero's rule was 13 years, 8 months which would be June 11th
Suetonius claims that Nero committed suicide in Suetonius, The Lives of Twelve Caesars, Life of Nero 49; Sulpicius Severus, who possibly used Tacitus' lost fragments as a source, reports that is was uncertain whether Nero committed suicide, Sulpicius Severus, Chronica II.29, also see T.D. Barnes, "The Fragments of Tacitus' Histories", Classical Philology (1977), p.228
